THE ELASTIC SCATTERING OF Co60 y-RAYS1

I<. G. STANDING AND J . V. JOVXNOVICI-I? Departtilent of Physics, University of Manitoba, Winni f ieg

Received April 24, 1961

ABSTRACT

Cross sections have been lneasured for 90" elastic scattering of Coca 7-rays (1.33 and 1.17 Mev) from aluminum, copper, and tin. They agree with theoretical predictions for Rayleigh and nuclear Thomson scattering. A continu~um between the Colnpton and the elastically scattered -/-rays interfered with the measure- ments for heavier elements, since i t increased rapidly with Z.

The continuun~ was investigated by observing thc ?-rays scattered through 60" by thin gold foils. I t was found to be produced directly by the ?-rays them- selves, not by the electrons they 1;nocl; out of the target atoms. The expcrirnental cross section was compared with that expected for incoherent scattering from the bound atomic electrons. I t agrees within a factor of 2 with the cross section given by the form factor approximation, although it is more than an order of magnitude larger than a modification of the form factor, which is based on cornparison with elastic scattering calculations.

The spectrum of elastically scattered ?-rays from lead was examined by a special method in order to reduce the contribution of the inelastic continuum; two photomultipliers faced a single scintillator and pulses from one photo~n~llt i- plier were counter1 only when they corresponded to the largest pulses from the other. The cross section between 12" and 150" mas found to be much lower than that of luost previous measurements. I t agrees with thc calculated cross section for Rayleigh and nuclear Thomson scattering within the possible errors in the calculations. Therefore the measurements yield no evidence for the presence of Delbrilcli scattering, the upper limit on its intensity now being set primarily by the uncertainties in the theory.

A. INTRODUCTION

The elastic scattering of y-rays of energy -1 Mev is of interest as a nleans of investigating Delbruclc scattering, the scattering of photons by a static electric field (for an extensive discussion, see Jauch and Rohrlich 1955). This is a process closely related to the scattering of light by light, both effects being predicted by quantum electrodynanlics as a result of the polarizatioil of the vacuum. Vacuum polarization accounts for part of the Lamb shift, and the excellent agreenleilt of theory and experiment there is comtl~only talien as verification of its existence, but Delbrucli scattering could provide an inde- pendent checli on the theory.

The interpretation of experiillental results on Delbriick scattering has been con~plicated by the difficulty of disentangling it froin various other processes which occur when y-rays interact with matter. Possible priiuary effects inay be classified (Fano 1953) by the liind of interaction responsible (with nucleons, with atomic electrons, with the electric fields around charged particles, or with the meson fields around nucleons,) and by its result (absorption, incoher- ent scattering, or coherent scattering). In addition to these there may be secondary effects caused by the electrons produced when y-rays are absorbed or incoherently scattered.

'A suln~nary of this worlc has already been published (Standing and Jovanovich 1958). leave from the University of Belgrade; now a t the Department of Physics, IVashington

University, St. Louis, NIo., U.S.A. Canad~an Journal of Physics. Volume 40 (1962)

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STANDIXG A N D JOVANOVICH: COBALT-GO -/-RAYS 623

Here coherent scattering nleans scattering in which the i~lternal state of the atom is unchanged, so contributions to the scattering fro111 different parts of the same atom co~nbine coherently (by addition of amplitudes). Because the internal atomic state is unchanged, the y-ray recoil must be talten up by the atoll1 as a whole, so the scattering is approximately elastic. This provides a lnealls of distinguishing the coherent scattering fro111 the much more probable incohere~lt Coinpton scattering, and fro111 other inelastic effects. Experiment- ally this separation is the chief problem.

Even if the elastic scattering can be isolated, however, there remain diffi- culties in its interpretation, since it may arise froin several sources. Corre- sponding to the classification above, scattering from a given atoll1 inay include coherent co~ltributions fro111 the nucleons, from the ato~nic electrons, from the nuclear and electronic electric fields, and froin the nuclear ineson field. Nuclear resonant scattering a t energies ~1 Mev has been estimated as sillall (Levinger 1951), so the contributions of the nucleons should be given silnply by the classical Thornson scattering fro111 the nucleus. Electronic electric fields should contribute a negligible amount compared with the nuclear electric field, and ineson effects should likewise be negligible a t this energy (Toll 1952). Thus we are left with contributions to the coherent scattering fro111 the nucleons (nuclear Thomson scattering), fro111 the bound electrons (Rayleigh scattering), and fro111 the nuclear electric field (Delbriiclr scattering).

In order to identify Delbriiclc scattering, i t is therefore necessary to know the theoretical cross sections for nuclear Thomson scattering and for Rayleigh scattering, the for~ner being well known. Rayleigh scattering was calculated in the form factor approxilnation (Franz 1935; Levinger 1952; Nelins and Oppenheiin 1955), which neglects the effect of the nuclear Coulon~b field in the intermediate states of the electron, until Brown and Mayers (1957) carried out a proper calculation which tooli this into account. They calculated the Rayleigh scattering fro111 the mercury I< shell of y-rays with energies up to 2.56 mc2, where the scattering cross section was much sillaller than the for111 factor result, by allnost an order of magnitude a t some angles.

Accurate calculations of Delbriick scattering itself have been carried out only for a scattering angle of 0" (Rohrlich and Gluckstern 1952; Toll 1952), where its amplitude is about five times the amplitude for nuclear Thomson scattering, and it is out of phase with the Rayleigh and Thoinson scattering. The scattering amplitude consists of two parts, an "absorptive" (imaginary) part, which is directly related to ordinary pair production, and a "dispersive" (real) part, which is the part arising froin vacuuin polarization, and is there- fore more interesting. Below the pair production threshold the "absorptive" part is zero, but a t higher energies it increases more rapidly a t O0 than the "dispersive" part and predominates above about 10 Mev. For this reasoil i t seems most valuable to look for Delbriick scattering a t energies of a few Mev or less, where the dispersive part is expected to predominate.

Experin~e~lts looliing for hard conlponents in scattered ?-radiation were first done nlally years ago, and the early work has been reviewed by Storruste (1951). However, the first experiinents to indicate the possible presence of

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624 CANADIAN JOURKAL O F PHYSICS. VOL. -10, 1962

Delbruck scattering were those of Wilson (1953), who measured the scattering of 1.33-Mev y-rays from lead. His ineasured cross section was much smaller than the cross section calculated for Thomson and Rayleigh scattering using the for111 factor. I t was argued that since the experimental errors mere such as to nlalte the observed cross section larger than the true cross section, and since i t seemed unliltely that the forin factor calculations were bad enough to account for the discrepancy, i t could probably be blamed on Delbruclr scat- tering. This would require Delbruck scattering to interfere destructively with the nuclear Thomson and Rayleigh contributions, a reasonable result since destructive interference is expected theoretically a t least for 0". More recent experiments have indicated similar conclusions, but the accurate calculations of Rayleigh scattering have completely changed the interpretation of the experimental results. The results now lie above the values predicted by nuclear Tho~nson and Rayleigh scattering, rather than below them, raising the question whether the difference is caused by Delbruck scattering or simply by experi- mental errors. The latest published results a t 1.33 Mev (Hara et al. 1958; Bernstein and Mann 1958) have quoted errors -20% but differ froin one another by factors -2 a t some angles, indicating that the experimental situation is still far from clear.

Preliminary work (Hardie 1956; Jovanovich 1956) showed that considerable improvements in experimental accuracy might be expected from the use of a different method of examining the y-ray spectrum, and that the commonly accepted explanation of the inelastic baclcground was not valid. We decided to worlt a t 1.33 Mev since theoretical interpretation of the results was more straightforward there than a t higher energies, and i t seemed reasonable to clarify the experimental situatioil there first. In order to untangle the various processes observed, we ineasured the elastic scattering by two different methods and did a separate experiment to obtain information on inelastic scattering. I t is thus convenient to separate this report into several parts: Part B includes the apparatus, theory, and measurements that are common to all three experiments; Part C deals with the experiments on elastic scattering measured in a conventional way; Part D treats the inelastic scattering; Part E deals with the elastic scattering measurenlents using the "two-photomulti- plier method"; and Part F contains the discussion and conclusions.

B. GENERAL MEASUREMENTS

( 1 ) Apparatus Figure 1 shows a typical view of the experimental apparatus. A Co60 y-ray

source of strength -1500 curies was enclosed in a beam therapy unit* whose aperture was normally closed by a reservoir of mercury. When the mercury was blown out of the reservoir, CoGO y-rays were free to strilte the target, and the scattered y-rays were observed by a NaI(T1) crystal in a scintillation counter. The counter with its shielding was mounted on a table which could be moved to different scattering angles. At each angle enough lead absorber was put between the target and the counter to malce pileup of inelastically

*Manufactured by Atomic Energy of Canada Limited, Ottawa, Ontario.

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STANDING A N D JOVANOVICH: COBALT-60 7-RAYS

S O U R C E

M E R C U R Y S H U T T E R

0 10 20 30 cm - SCALE

FIG. 1. Cross section of the apparatus (median plane) as seen from above. The bean1 therapy unit included the source with its surrounding shielding, a reservoir full of mercury which could be blown out by compressed air to allow the ?-ray beam to escape, and variable slits (horizontal and vertical). The rectangular target was hung by threads from a support above. The thickness of the lead absorber in front of the counter was =1.5 cm for B = 150°, ~ 2 . 5 cm for B = 120°, ~ 4 . 5 cm for B = 90°, = G cm for B = 60°, =8 crn for B =-3O0, and =9 cm for B = 12'.

scattered y-rays negligible in the region of 1.33 Mev. The target was placed so that the angle of scattering fro111 all points on its surface was the same to first order-it is easily shown that this requires sin @/sin ( 0 4 ) = r/R where r is the source-target distance and R is the counter-target distance.

The y-ray beam from the source was limited first by a square aperture in front of the mercury reservoir. The beam could be further limited by the variable slits which gave a rectangular aperture variable in both dimensions. T o cut down radiation in the room, these slits were generally adjusted so that the y-ray beam illuininated the target area only (plus a generous allow- ance for error), the size of the target itself determining the effective bean1 size.

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6% CANADIAN JOURNAL O F PI-IYSICS. VOL. 40, 1962

Adjustn~eilt of the slits and positioni~lg of the target were done with the aid of an internal light in the beam unit, reflected from a mirror so that its image coincided with the source position.

Because of the large amount of scattered radiation in the room, i t was necessary to be rather careful about the shielding around the counter. About 1200 Ib of lead were required. Radiation scattered from the beam unit slits into the counter was particularly troublesome, so the shielding between the slits and the counter had to be rearranged a t each angle. However, i t was found possible to reduce the background without target so that in the neigh- borhood of 1.33 Mev i t was independent of whether the inercury shutter was open or not. Background a t lower energies arose ~nostly from air scattering, ,

and was not serious in the measurement of the elastic cross section. The electronic apparatus was conventional. Pulses froin the scintillation

counter were amplified by linear amplifiers and then fed into a 5-channel kicltsorter (Cool;e-Yarborough et al. 1950). For counting rates beyond the capacity of this instrument, a single-channel analyzer and a fast scaler were used.

(2) iMeas~irenzent of Cross Sections A simplifying feature of y-ray elastic scattering cross sections is that they

can be measured absolutely by two relative measurements. T o see how this comes about, consider the counting rate in the counter from the elastically scattered radiation :

where na is the number of counts per second registered in the counter, a is the number of 1.33-Mev y-rays per second emerging from the beam therapy unit Co60 source, r is the source-target distance, N is the number of target atoms, (dg/dQ),, is the differential elastic scattering cross section, w is the solid angle subtended by the counter a t the target, and €a is the probability of a y-ray elastically scattered within the solid angle w being registered by the counter.

Now suppose the y-ray beam is shut off, the target removed, and a small CoGO source spread over the same area as the target (b 1.33-Mev y-rays per second) is placed in the target position. Under these conditions the number of counts registered by the counter is:

where ~b is the probability of a y-ray emitted within the solid angle w being registered by the counter. Since Ea applied to elastic scattering, c = EL, = E, so from eqs. (1) and (2)

Therefore, besides the number of atoms in the target and the source-target distance, i t is necessary to know only the ratios na/nb and b /a to obtain the

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STANDING AND JOVANOVICEI: COBALT-GO ?-RAYS 627

absolute differential cross section. The next section describes the measurement of b/a, and the measurements of n,/?zb are given in the later parts.

( 3 ) The Ratio b/a In principle the ratio b/a is sinlple to determine by a direct comparison

of the counting rates in a counter from the two sources in the same geometry. Here, however, the very small size of the ratio made a direct measurement impractical, and the limited size of the beam unit room pre- vented the use of the inverse square law. The difficulty was circumvented to some extent by first comparing the source b with a CoGO source of inter- mediate strength d disintegrations per second (-150 mc), then comparing source d \\rith source a. I n practice it was found convenient t o use a second intermediate source (-1.5 n ~ c ) a s well and t o d o the first measurement in two steps. This was straightforward, direct comparison of the counting rates in a distant counter with the sources in the same position giving b/d = 1.37 x 104.

T h e ratio d/a was measured b y placing a well-shielded scintillation counter in the direct beam from the beam unit 230 cm from its source. A collimator extended fro111 70 cm to 100 cm from the source, and lead absorbers were inserted behind the collimator. Absorption curves were taken for the beanI unit source a, then for source d when the beam unit was moved ou t of position

NUMBER O F ABSORBERS

FIG. 2. Absorption curves for Co60 source d (-150 mc) and a (bean1 unit). Each absorber was ~ 4 . 0 g/cm2 thick. The absorption curve A for the beam unit source was limited by ex- cessive counting rates to >/ 38 absorbers; the curve B for the intermediate source was limited by background to < 43 absorbers. The lines are straight lines calculated for a least squares fit to the experimental points. They have the same slope, corresponding to an absorption coefficient = .054 cm2/g, in agreement with Colgate's measurement (Colgate 1952) corrected for solid angle.

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628 C:\NXDIA?: JOURNAI, OF PI-IYSICS. VOL. 40, 1963

and source d was placed a t the previous location of the beam unit source. They are shown in Fig. 2. I11 this measurement, as in the previous ones, only y-rays in a narrow energy range (- 3 ~ 5 7 ~ ) a t 1.33 Mev were counted. Measure- ineilts were made for several different sizes of collimator aperture to ensure that the collimator was not intercepting y-rays heading for the counter; how- ever, in each case the two sources were compared with the same collimator aperture, an important point, since the change in absorption \\rith size of collinlator (because of small angle scattering from the absorber into the counter) is significant for these large absorber thicltnesses. The absorption is exponential within the errors, and the ratio of the two curves gives d /a = 0.97 X lo-", malting the over-all ratio b /a = 1.33 X

If we talce all the poiilts on curve B into account, the calc~~lated statistical error is very small. However, this calculated accuracy comes mostly from points a t small absorber thiclcness, and systematic errors may arise in extra- polating to large thiclcness. Therefore we have also fitted a line to the six experimental poiilts closest to the end point of B. Here the cornpal-ison with A is direct, the result agrees with the one for the whole curve quoted above, and the standard deviation is -&YO. between different measure- ments of the ratio were within this range, and we adopt &5% as our estin~ate of the probable error in b/a.

C. ELASTIC SCATTERING: CONVENTIONAL METHOD

Figure 3 shows the pulse-height distribution obtained for 00" scattering fro111 various target elements, using a conventional scintillation counter as detector. The spectra all show a peak a t low energy, structure a t the full energy which inay be identified as elastic scattering, and a continuum be- tween. The low-energy part of the scattered spectr~un~ has been greatly depressed by the lead absorber between the counter and the target. This effect is 1110st apparent for the Compton-scattered radiation, which for zero absorber gives a peak a t -8 v several orders of magnitude higher than the present spectra; here the absorber has attenuated it by a factor -1O"elative to the elastically scattered radiation, so that it is not even visible in the spectrum.

The low-energy peak which does appear in all the spectra on top of the contiiluuin comes from multiple Compton scattering (Storruste 1051) and froin annihilation radiation. These effects are less probable by more than an order of magnitude than single Conlpton scattering, but their larger energy leads to much slnaller absorption; e.g. two successive Compton scatterings in the target, each through 45O, give a final y-ray of energy -4/3 the energy of a y-ray after a single Compton scattering through 90°, and the doubly scattered y-ray is attenuated by a factor -200 less than the singly scattered y-ray. For the aluminum target the peak comes mostly from doubly scattered y-rays and the observed cross section is in agreement with rough calculatioils for this process. For the lead target the peak comes mostly from annihilation radiation from positrons produced by the y-rays in the target.

As discussed above, the elastic scattering cross section is determined by

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STANIIIXG A N D JOVANOVICII: COB.4LT-GO ?-RAYS

PULSE HEIGHT (VOLTS1

FIG. 3. Pulse-height distributions from CoGo 7-rays scattered through an angle of 90" from various target elements (Al, 6.7 g/cm2; Cu, 2.3 g/cm2; Sn, 1.4 g/cm2; and Pb, 0.8 g/cm?), as observed in a NaI(T1) crystal (2.5 in. diam. X2.75 in. long) viewed by a Dumont 3 in. diam. photomultiplier. Targets were ~ 5 . 5 in.X6 in. and the source-target and counter-target dis- tances were =90 cm and =70 cm. The dotted curves indicate the spectrunl of a CoGo source placed a t the target position, and show the characteristic pealts corresponding to the 1.33- and 1.17-hlev ?-rays; they are normalized to the 1.33-Mev peak in the spectra of 7-rays scattered from A1 and Cu. Here the scattered spectra from A1 and Cu were taken with 60 g/cni2 Pb absorber in front of the counter, and the spectra for P b and Sn with 56 g/cm2 Pb absorber, and the spectra are corrected for target absorption only a t 1.33 Mev, so the curves are not directly comparable. Pileup can account for only a small part of the continuum in A1 and is negligible for the heavier elements, since their absolute counting rate is smaller. '

the ratio of source strengths b/a and the ratio n a / n b of the counting rate from elastic scattering to the counting rate fro111 the small Corn source. The latter ratio can be obtained directly frorn the data shown in Fig. 3 if there is no intruding baclcground under the elastic peaks. For alu~llinunl the elastic

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G30 CANADI.AN JOURNAL OF PI-IYSICS. VOL. -10. 1962

peaks are good replicas of the peaks in the CoGO spectrum, so the background lnust be small, but with increasing Z the scattered spectra becoine more and Illore distorted. Part of the distortion is caused by the larger cross section for Rayleigh scattering of the lower-energl~ y-ray a t 1.17 Mev, but clearly there may also be intruding background froill the continuum. Before investi- gating the source of the continuunl it is hazardous to n1al;e assuillptions about its contribution, but a t least i t seems reasonable to assume that i t is a sinootll function of energy. The ratio n,/nt, was therefore found by subtracting various fractions of the source spectruin nb froin the scattered spectrum n, and requiring the resulting points to lie on a snlooth curve. Coinbinatioil of the results so obtained with the ratio b/a yielded the cross sections shown in Table I ; the errors indicate the range of values of n,/nb which will give a

TABLE I

Cross section for elastic scattering of 1.33-i\/Iev ?-rays a t 90°

Theoretical scattering amplitude (units of ro)

Theoretical Esperimen tal Yuclear cross section cross section

Thornson* Ravle i~h* (cm2/sterad) (cm2/sterad)

Al .0017 . 0000 1 0.4G X 10PO (0. -19 f .04) X Cu ,0036 .00028 2 .2 ~ 1 0 - 3 0 (2.0 +.5) X10-30 Sn ,0058 ,0032 9 . 1 X10-30 (8.8 f 1.5)X10-30 Pb .0089 ,039 12GX (10G+14) X10-30

*This is the nuclear Tliornson amplitude for both spin flip and no spin flip. T h e Ilayleigll scattering amplitude given is the form factor for spin flip, \ \ r l~ ic l~ the calculations of Brown and Mayers (1967) show to be a good approximation. They also s l ~ o w tha t the no spin flip amplitude is small, so me have neglected i t escept for the case of leacl, where \re liave used tlieir calculations wit11 corrections for Z and energy.

*This is the nuclear Tliornson amplitude for both spin flip and no spin flip. T h e Ilayleigll scattering amplitude given is the form factor for spin flip, \ \ r l~ ic l~ the calculations of Brown and Mayers (1967) show to be a good approximation. They also s l ~ o w tha t the no spin flip amplitude is small, so me have neglected i t escept for the case of leacl, where \re liave used tlieir calculations wit11 corrections for Z and energy.

smooth bacliground curve (cf. Fig. 4). Also shown are the corresponding theoretical cross sections.

As Z increases, the uilcertaillty iilvolved in the subtractioil of the bacli- ground froin the coiltinuuill beconles inore serious until i t is the chief liinita- tion to the accuracy of the elastic cross section. RiIoreover, a t angles of scatter- ing sinaller than 90' the situation is even worse; for example, the spectrum of radiation scattered from lead a t GO0 is shown in Fig. 4. Here the observed elastic peak inay be fitted eve11 when one-third of it is assumed to be back- ground, we so estiinate n,/nb = .075 with an error of a t least f 20y0.

At scattering angles >90° this uncertainty in bacligrouild subtraction is not serious, but for sinaller angles it is the chief problem. The origin of the bacligrouild continuum will be discussed in Part Dl and in Part E a method of dealing with it will be described. However, results for the elastic scattering cross section from lead were obtained using the coilventional method described here (within the limitations described above) and are given in Table I11 together with the inore accurate results of Part E.

D. INELASTIC SCATTEIiING

( 1 ) The Sol~rce of the Continz~z~m I t is clearly important to find the origin of the continuum in order to inter-

pret the elastic scattering, as well as for its ow11 sake. Obviously it could arise i f small pulses were piling up on top of one another in the amplifier; however,

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STAXDING A N D JOV.4NOVICI-I: COB.-\LT-60 -.-R.\YS

P U L S E H E I G H T (Volts)

1 . 4. Scintillation counter spectrum of Coca ?-rays scattered through an angle of 60" from lead (0.8 g /c~u?) . The arro\vs indicate the positions of the two peaks in the Co"spectr~~m. The curve a t the left is the scattered spectrum itself, and above i t is sho\\rn the spectrunl observed from the CoGO source b a t the target position (on a n arbitrary scale). The two curves on the right show the result of subtracting 9% and 6% of the source b from the scattered spectrum. They represent reasonable upper and lower linlits on the amount of the 1.33-Mev peak which may be s~~b t rac t ed from the scattered spectrum to yield a smooth curve in that region. As expected theoretically from the larger Rayleigh cross section, the scattered intensity of the l.l 'i->lev ?-ray is higher than that of the 1.33-Mev y-ray.

this effect varies roughly as the square of the counting rate and even for aluminum, which has the largest absolute counting rate in the data of Fig. 3, it accounts for only a few per cent of the continuum, so that for the other ele~nents it illust be negligible. Multiple Compton scattering in the target gives the low-energy peaks in aluminum but is negligible a t higher energy.

An explanation of the continuum, which has often been advanced (see, for example, Messelt and Storruste (1956) and Goldzahl and Eberhard (1957)), ascribes it to a secondary effect of the photoelectrons, Compton electrons, and pairs produced in the target by the incident 7-rays. In collisions with the target atoms the photoelectrons and Coinpton electrons may give off breins- strahlung, and the positrons may annihilate in flight. Either of these events gives a continuum of y-rays with maximunl energy close to the full y-ray energy, and has a high Z dependence.

If these secondary processes are responsible for the continuum, it should be possible to reduce its intensity greatly by using a target thickness which is small compared with the range of the secondary electrons. In that case most of the electrons would escape from the target and lose their energy in air, where the probability for bremsstrahlung or annihilation in flight is small. Preliminary work (Jovanovich 1956) had shown that for lead, tin, and copper

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632 C.AX.-1~1.1~ JOURS.II~ OF PIIYSICS. VOL. -10, 1302

nearly all of the continuum remained even when the target thickness was reduced to less tllan lOy0 of the range of a 1-Mev electron. The data was completely inconsistent with an origin in secondary processes, a ~ i d so pointed to a primary process, which of course would be i~idepeildent of target thicl;- ness. A similar experiment has been reported by Goldzahl and Banaigs (1958).

A possible prinlary process is Compton scattering from the bound electrons of the atom, an effect well 1;nown for X rays (Dumond 1933; Evans 1958). M7hen a photon scatters from a free electron a t rest, its losses in ~no~nentulll and energy, which are given entirely to the recoil electron, depend only on the angle of scattering. If the electron is originally bound within an atom, however, ~no~nentum may be given to the ionized atom which it leaves behind; consequently the momentum and energy of the scattered photon may differ from those determined for scattering from a free electron. The energy of the scattered photon may in fact have any value up to its original energy less the electron binding energ),; there the ionizecl atom takes all the recoil momentum and the electron is set free with zero velocity.

If the electron binding energy is very much smaller than the free electron recoil energy, the momellturn talcen by the ionized atom will usually be small, and under these two co~lditions the binding may be approximated simply by its influence on tlle electron monientum (Burl~hardt 193G) ; i.e. the scattering is calculated as i f the electrons were free, but possessed the sallle momentum distribution that they have within the atom. 111 this approxilnatioil the photon scatters from a ~noving free electron of momentum p so the ionized atom must recoil with a momentum -p, and the change in photon frequency may be pictured as a Doppler shift caused by reflection fro111 a moving target (Dumond 1933). The approximation accounts fairly well for the gross features observed in scattered X-ray spectra (Burl<hardt 1936; Duncanson and Coulson 1943).

Our measuremellts are particularly co~lcer~led with the scattered y-ray spectruni near its high-energy end point, where inost of the y-ray lnolnentu~ll loss is given to the ionized atom, so the above approximation is no longer valid. In spite of that , the preliminary results (Jova~~ovich 1956) agreed with its predictions within an order of magnitude. Meanwhile the scattered spcc- trum had been calculated without assuming a small atomic recoil (Randles 1957), so it seemed worthwhile to carry out better measurements.

(2 ) ilfeaszirement o j the Inelastic Scattering The experiment was performed in almost the same way and a t about the

same time as the one on elastic scattering described in Part C. The only changes were in the size and thickness of the targets, the beam cross section, and the distance between the target and tlle counter (R = 82.1 cm). In order to eliminate as nluch as possible the influence of the secondary effects mentioned earlier and to show experimentally that they are llegligible in conlparison with primary effects, we used three very thin gold targets No. I , No. 2, and No. 3; they had thicknesses of 23.98 mg/cm2 (0.0005 in.), 47.64 mg /c~~ l~O.OOl in.), and 95.68 1ng/cm2 (0.002 in.) respectively. A I-Mev electron loses about 48 kev of energy (Nelms 1956) in traversing perpendicularly a 0.001-in. thick foil of gold, so tlle effective thickness of all the gold foils used was much s~naller than the range of the electrons.

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STANDIXG A N D JOVANOVICI-I: COBALT-60 -/-RAYS 633

The measured pulse-height spectra are shown in Fig. 5. The ratio of counting rates per unit of inass in the pulse-height range between 17.5 v and 24 v for targets No. 3 and No. 1 is 1.01, and for targets No. 2 and No. 1 is 0.94; the

lo3

lo2

w 10 I- 3

Z I \ I a @= (I

\ I Y) I- z 3 0 0

0.1

5 10 15 20 25 30 15 20 25 30 PULSE HEIGHT (Volts)

FIG. 5. I n the left-hand part of the figure are shown the pulse-height spectra of CoGO ?-rays scattered through 60" fro111 gold targets No. 1 and S o . 3 (each 3 in. sq.), normalized to unit weight. The results for target No. 2 were omitted to avoid overcrowding of points. The heavy linc (cur1.c :\) is the theoretical pr~lse-height spectrum computed by fitting eq. (S), i.e. by assumil~g the incoherent spectrunl shape given by the form factor, and nor~nalizing i t t o give the best fit to the observed spectrun~. On the right the full circles give the scattered spectrum averaged over all three targets. The solid line B represents the pulse-height spectrum com- puted from the incoherent and coherent cross sections obtained by a trial and error procedure, i.e., from curve C of Fig. 6. The dashed line is the part of this due to incoherent scattering.

deviations froin unity are believed to be caused by drifts in electronics and other small systematic errors. If secondary effects were important they should appear as a contribution to the pulse-height spectrum proportional, in first approximation, to the target thickness, so i t is clear tha t such a contribution must be a few per cent of the continuum or less. Also this shows directly that very few pulses are piling up.

We therefore conclude that allnost all of the continuum must be caused by primary effects. The paucity of secoildary radiation is understandable, since one would expect bremsstrahlung from Compton electrons and photo- electrons to be its most important source, and a crude but fairly conservative estimate of the brenlsstrahlung gives a figure about two orders of magnitude smaller than the continuum we observe.

(3) Calculation of the Incoherent Scattering Cross Section As mentioned earlier, the primary effect that seeins most lilcely to produce

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634 CAN.ADIAN JOURNhL O F PHYSICS. VOL. 40. 1962

the observed continuun~ is inelastic scattering from the bound atomic elec- trons. The relativistic cross section for incoherent scattering fro111 the electrons of the K shell has been derived by Randles (1957); we shall refer to his paper as R. He used correct wave functions for the initial and final atonlic states, so his calculatioils are not restricted to small atomic recoil, as are the free electron ones, but he did neglect the effect of binding in the internlediate atomic states. This corresponds to the forin factor approximation for coherent scattering, so we shall call his calculation by that name also. As mentioned earlier, the form factor approxin~ation is very poor for calculating the coherent scattering (Brown and Mayers 1957) and the same might be expected for incoherent scattering. However, Randles (R $7) has argued tha t the ratio of incoherent to coherent cross sections, both calculated in the fornz factor approximation, should be approxinlately the same as in the correct theory; i.e. the ratio of the true incoherent cross section to the cross section calculated in the forin factor approxi~nation should be the same as the ratio of the coherent cross section calculated by Brown and Mayers (1957) to the corre- sponding form factor cross section.

The for111 factor cross section for incoherent scattering fro111 both electrons of the K shell, as given in (R $8),* is

q(l+cos20) 2 ~ " ' dk' = r i --

2 T k p (TV+ l)e"Z"'"lP

form factor dk'

where m = c = f i = 1, k is the incident and k' the scattered photon energy, ro the classical electroil radius, Z the charge of the nucleus, a the fine structure constant, Ak = (k"k'2-2kk' cos 0)'12 is the change of momentum in inco- herent scattering, and p = [(k - k ' + y ~ ) ~ - 1]lI2 is the n~omentum of the ejected electron. The other symbols are defined in R (eqs. (8.3) and (8.4)).

In order to compare with experiment, the cross section was co~nputed from eq. (4) using the IBM 650 automatic digital computer of \Vashington Uni- versity, St. Louis, Missouri, for gold (2 = 79), a scattering angle of GO0, incident photon energies kl = 1.332 Mev and kz = 1.172 Mev, and for six different values of momentum p of the ejected electrons. I t is estimated that the over-all numerical accuracy is 1% or better for the srnallest values of p (largest values of k'), and 10% or better for the largest values of 9. The computed cross section is shown in Fig. 6 for the two incident photon energies (curves A and B). We have also calculated the cross section near the end point as a function of angle and have plotted the result in Fig. 7.

*Our eq. (4) is equivalent to eqs. (8.1) ancl(8.2) of R with the integral and weighing function omitted, the only dillerence being that in R (eq. (8.1)) the incoherent cross section is given as a function of the momentum of the ejected electrons and in our equation as a f ~ ~ n c t i o n of the scattered photon energy. Errors in R which were kindly pointed out to us by Mr. Randles have been corrected in our expressions: in his (8.2) the factor 2"(X)-2 should read (2p)"(")-2 and there is a factor of rr missing from the denominator; in eq. (8.4), [ (Ya- l ) . . . ]'12should read [(1 - ? o ) . . . ]u2. 111 R , eq. (4) was expanded in powers of p and integrated overa weighted energy interval near the end point, thus restricting its validity to small values of p; t o renlove the restriction we have computed the cross section from it directly.

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FIG. 6. Cur\,es X and B are the theoretical cross sections computed from eq. (4) for in- coherent scattering through 00" from the K electrons of gold a t incident photon energies of 1.33 Mev and 1.17 RIev respectively. The cross section, here denoted by gI, is [(&/an) (k, k')Jr,,,, r,,t,, in the notation of the text. The full circles connected by curve C represent the cross section obtained by a trial and error procedure to fit the experimental data; contrib~~tions from the two incident energies have not been resolved.

I . 7. The curves give the variation with arlgle of the theoretical incoherent scattering cross section (form factor) for gold, 1.33-hIev incident ?-ray energy, and for several values of the m o m e n t ~ ~ m p of the ejected electron (units of mc). The corresponding curves for lead sho~lld be about 4 0 as large. The closed circles are the experiniental cross sections for lead near thc end point (see Section E.2).

(4) JIeasurenzent of Cross Sections Generalizing eq. (1) to talte account explicitly of the inelastic scattering,

the two y-rays of CoGO, and the continuous distribution of pulses from the counter, we obtain :

+ J.(kl)R(kllh5) ( g ( k l k l ) ) lnel dkl

+ two similar terms for incident energy kz) I where kl (1.33 Mev) and kr (1.17 Mev) are the incident y-ray energies, k: and kI are the correspondiilg y-ray energies after inelastic scattering, ma(hj) is the couiltiilg rate in channel j of the lticltsorter (pulse height h,), a(k)R(k,h,)

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G3G CAxADIAN JOURNAL O F PI-IYSICS. VOL. 40. 1962

is the probability of a y-ray of energy k within the solid angle w being regis- tered in channel j of the lticltsorter, [(du/dQ)(k,k')llneldk' is the differential cross section for inelastic scattering of a y-ray of energy k into the energy range dk', and the other syinbols have been defined in Sectioil B(2).

Similarly, eq. (2) for the spectruin froin the calibrating source becomes:

(6) bw

mb(hj) = -[~(kl)R(kl,hj) +~(k?)R(k~,h j ) l . 43r

As before, dividing (4) by (5) yields:

ma(h') *(l+Ps) = .I+ S p i j (%(k1,k;)) dk: (7) n b ~ ah7 lnel

+Pzj~+ JP;~ (~(k2 .k : ) ) lnel dk:

where

This equation is basically the same as eq. (3) and in addition to the ratios required there, we need only to know the relative values P,, of the counting probabilities e(k)R(k,hj) for our energy range in order to find the absolute cross sections.

To obtain this ratio Pi, of counting probabilities for different energies, we measured the pulse-height spectra of CoGO (1.33 and 1.17 i\/Iev), Ka22 (1.28 and 0.51 Mev), ScIG (1.11 and 0.89 iVIev), and ZnG5 (1.11 Mev) from sources placed a t the target position with the sanle geoinetry and absorber thickness as in the scattering ineasurerneilts above. The coinplex spectra of the first three were resolved into colnpoilents with the aid of the ZnGs spectrum, and since the relative abundance of their two y-ray lines is well l;no\vn in each case, determination of the relative coullting probabilities was straightforward. The relative countiilg probabilities for interinediate energies were obtained by iilterpolation with an estimated accuracy of 20y0 or better in each point.

Although in principle it should be possible to calculate both the elastic and inelastic cross sections directly from eq. (7) and the observed coulltiilg rates, in practice the computation involves difficulties and ambiguities for a con- tinuous spectruin (Dixon and Aitlten 1958), particularly with two y-rays present. Therefore we followed the reverse procedure, calculating the pulse- height spectrum predicted by theory and comparing i t with the experimental spectrun~. If the shape of the incoherent scattering spectruin is given by the form factor approximation, i.e., i f

(g(kjkr)),e, = a ( s ( k , k ' ) ) form tnctor

eq. (7) becomes

(8) ma (h $) br

--(l+P2j) = al+C13ff1+P?,a2+C2,cr, mb(hj) aN

where Cij = Jpij ($(k,,k:)) form fnctor dk:

may be calculated from the measured P& and the con~puted form factor cross sections. The other constants in eq. (8) are known from the measure-

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STAXDING AND JOVANOVICFI: COBALT-GO 7-RAYS 637

ments described earlier, so we are left with variables u1, u2, al, and as, which nlay be fitted to the observed spectruln m,(h,) over a range of channels j and conlpared with the values predicted by theory. If the for111 factor gives the correct incoherent scattering cross section, we would have of course al = a? = 1. On the other hand, if Randles argument (R $7) is valid, then al and a2 should be equal to the ratios of the coherent cross sections in the correct theory to those in the for111 factor approxin~ation (-0.1).

The best fit of eq. (8) to the observed spectrum between 21 and 33.5 volts is shown in curve A of Fig. 5. Here a1 = 1 .7 f 0.4 and u1 = 0.20f0.03 inb/sterad, but only rough estiinates for the 1.17 line could be made because of interfer- ence from the 1.33-Mev inelastic spectrunl (a1 - 0.4 and u1- 0.3 mb/sterad). Curve A falls below the experimental points a t the lower end of the spectrum and attenlpts to raise it by fitting eq. (8) to a wider range of points gave curves which were too high at the upper end.

Since the forin factor spectruln is therefore not quite steep enough, we calculated a spectrum to agree with experinle~lt by a trial and error procedure. I t gave ul = 0.19f 0.03 mb/sterad and the inelastic cross section shown in curve C of Fig. 6. The corresponding experinlental spectrunl is curve B of Fig. 5, which gives an excellent fit to the observed points. Contributions fro111 the scattering of degraded radiation fro111 the CoGO bean1 therapy unit should be subtracted fro111 this apparent inelastic cross section. From the calculations of Cormack and Johns (1958) we estimate that they account for -3% of the apparent cross section a t 1.2 NIev, -10yo a t 1 Mev, and -207, a t 0.7 Mev, although the exact correction depends considerably on the geometry. Including it, the over-all error in the inelastic cross section (curve C of Fig. 6) is estinlated to be -f 20% over nlost of the spectrum and -&50% a t the lower end (0.7 Mev) and in the high-energy tail.

(5) Comparison between Tlzeory and Experiment The 1.33-Mev elastic cross section u1 agrees within the errors with the

predicted theoretical value 0.1631 0.03 mb/sterad (cf. Part F) , calculated from Brown and Mayers' results (1957).

The esperinlental inelastic cross section near the end point has a1 = 1.73I0.4, so it agrees within a factor -2 with that predicted by the form factor for incoherent scattering from the K shell electrons. That is probably as good as might be espected fro111 the for111 factor approximation, especially since scattering from the L shell is not included in the calculation and would account for a t least part of the difference. Scattering froin the outer shells might also explain the cross section of Fig. 6 being steeper than predicted by the K shell calculation. On the other hand, the experimental data disagree strongly with Randles' proposal (R $7) that the incoherent scattering should have the same ratio to the form factor as the coherent scattering does a t the same angle; in fact, the observed cross section is larger than the form factor a t 60" while Randles would predict i t to be about an order of magnitude smaller. A similar situation holds in the scattering of 0.662-Mev y-rays (Jovano- vich and Hurley 1060, and private coinmunication).'

*Motz and Missoni ( 1 9 G l ) have also ~neasured incoherent scattering cross sections a t O.GG2 Mev but their results are integrated over photon energy, so they depend nlostly on the cross section near the Co~npton peal;.

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638 CASADIAN JOURNAL OF I'llYSICS. VOL. -10. 1962

We conclude that inost of the coiltiizuum is produced by a primary process rather than by secondary effects of the photoelectroils and Conzpton electrons. The cross section is consistent with that expected from incoherent scattering fro111 the bound atomic electrons if the for111 factor calculations are correct within a factor -2. If Randles' arguments are valid, illost of the coiltinuulll must come from some other primary process. However, his arguments are purely qualitative, so we believe that there is not much justification for assuming such a process unless accurate calculations of the incoherent scat- tering (similar to those of Brown and NIayers (1957) for cohereilt scattering) should disagree with experiment.

E. ELASTIC SCrZTTERING: T\T7O-PI-IOTOAIULTIPLIER METI-IOD

( 1 ) Experimental iVfethod As seen in the previous sections, the inelastic scattering is the inail1 limitation

to the accuracy of the elastic scattering cross-section measurements, since some of it is liltely to be included in the measured "elastic" scattering. Because it is a primary effect, i t cannot be removed by the use of compound targets (Goldzahl and Eberhard 1057) or similar devices. However, it seemed from preli~ninary experiments (Hardie 1956) that the use of a clifferent method of . . examining the sciiltillation counter spectruin (Standing and Peelle 1953) might give improved results. 111 this method two photomultipliers, No. 1 and No. 2, view the same scintillatil~g crystal, so that, in general, a flash ol light in the crystal will give pulses in both photonlultipliers. The "gated" pulse spectrum froin photomultiplier No. 1 is recorcled; i.e. a pulse from KO. 1 is recorded only if the correspo~ldi~lg pulse from NO. 2 is greater than a "gate voltage" V.

First co~lsicler a group of monoenergetic events in the crystal, e.g. mono- energetic y-rays being absorbed ill the crystal through the photoelectric effect. Recn~ise of fluctuations, either within the photomultipliers tl~emselves or in the amount of light reaching them, the pulses froill each photomultiplier will not have a constant Ileight, but will be distributed about some average value. If, on the one hand, the fluctuations occur entirely witl~in the photo- n~ultipliers, the distributions from the two pliotomultipliers will be uncorre- lated with each other. On this assumption, if the "gate voltage" TJ in photo- lllultiplier Xo. 2 is chaiiged, the only effect on the pulses recorded from pl~otom~~ltiplier No. 1 will be to alter their total number; the shape and positioll of the recorded spectra will renlain the same. If, on the other hand, the light is not divided in a constant ratio between the photomultipliers, or if there is a difference in the amount of light produced by different events in the crystal, the positioil of the recorded peak will change as the "gate voltage" is changed.

A crystal inounting, designed with the aim of dividing the light from an event in ally part of the crystal equally between the two photomultipliers, is shown in Fig. 8. Figure 9 shows the spectrum of the 1.28-Mev 7-ray from Na?? obtained with this mounting for various gate settings. Only a small shift in the position of the spectrunl (-2%) taltes place as the gate position is varied to give a reduction in counti~lg rate by a factor -1000, indicating

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ST.\SDISG .\XI3 JOVXSOVICII: COB.\LT-GO -,-RAYS

/=-ALUMINA COVERED WITH Al FOIL

'PHOTOMULTIPLIER NO-I PHOTOMULTIPLIER ~ 0 . 2 '

I 2 3 C M SCALE '-

FIG. 8. Crystal mounting. The XaI('T1) crystal was surrounded by lucite, the space between the crystal and the lucite and between the lucite and the photo~nultipliers being filled i ~ i t h silicone fluid (Dow-Corni~ig DC-200). The lucite was surrounded by a reflector of a-a lum~na, then by aluminum foil, and finally by all alumi~iurn housing sealed to the photomultipliersby vacuum was (Bell 1955).

that there is little correlation between the distributions from the two photo- multipliers. The fact that this shift is in the same directioil as the illovenlent of the gate seeins consistent with an intrinsic spread in light output froill the crystal (Kelley, Bell, and Lazar 1956).

Now consider two groups of inonoenergetic events of different energies. The effect of the gate will be the saine as before for each group but the intensity of the lower energy group will be cut more, and by raising the gate voltage high enough we may discriminate against i t by any factor desired. For esample, Fig. 10 illustrates the elimination of the 1.17-Mev y-ray spectrum in Co60 by raising the gate voltage.

By superposition, a similar result will hold for any number of lower energy groups, or for the continuous spectrum produced by inelastic scattering. The contribution of the continuum could be reduced in the conventional method also by going out far enough into the tail of the elastic peal;. There, however, the side of a steeply rising spectrum i-nust be measured, so it is extremely difficult to estimate background or shifts in peak position. In our nlethod they show up clearly in the experimental spectra. Since again a ratio is being measured, it is not necessary to know the absolute gate positions, but only to keep them the same for the measurement of n, and n b . The following section describes the results obtained in this way.

Another systematic error in the measurement of the elastic cross sectioil may be caused by a difference in counting rates in the ineasureinent of n, and nb. Since the elastically scattered radiation is accompanied by a much larger inelastic component, the counting rates will norinally be quite different, and it has been found (Bell, Davis, and Bernsteiil 1955) that a change in counting rate in a photomultiplier may lead to a change in its gain. Both in the convei~tioi~al method and in the one described above, this would lead to a reproducible error in cross section. The shift in gain depends in first approxi- mation only on the total pl~otomultiplier current, so we tried to keep it a s

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640 CANADIAX JOURNAL OF PIIYSICS. VOL. 40. 10G2

PULSE HEIGHT (VOLTS) PULSE HEIGHT ( V O L T S )

FIG. 9. Gated scintillation counter spectra of the 1.28 ?-ray fro111 Na2'. Here two photo- multipliers face a single crystal as shown in Fig. 8. The spectrum A is the normal ungated spectrum observed fro111 photomultiplier No. 1. B to G are the spectra of pulses from photo- multiplier No. 1 corresponding to pulses fro111 photomultiplier No. 2 greater than a "gate voltage" V as V is successively increased. To checlr that any shifts observed were real, an ungated spectrum was talren between each pair of gated spectra.

FIG. 10. Gated scintillation counter spectra of Corn. A is the ungated spectruln observed in photo~nultiplier No. 1 showing the two peaks corresponding to the 1.33-hilev and 1.17-Mev ?-rays. B to D are gated spectra observed in photo~nultiplier No. 1 as the gate voltage V is increased. \\Then V is increased beyond the value for D, the spectrum shows a behavior similar to that shown in Fig. 9, i.e. the 1.33-Mev pealr remains in approximately the same position and decreases in intensity.

constant as possible. During the measurement of n b the Co60 source was accoinpanied by a source of Cs137 placed a t such a distance fro111 the counter that the photomultiplier current was the sallle as during the measurement of n,. The background to nb was, of course, taken with the Cs137 present and the background to n, was taken with the beam striking an aluminu~n target to give again the same total current. In addition to this we were fortunate in having two photomultipliers which showed shifts in opposite directions; one had a gain which increased with current, and the other a gain which decreased, the first being a Du~noil t 6292 and the second an RCA 6342.*

"It has since been reported by Jung et al. (1960) that these two types of photomultipliers llorn~ally give opposite shifts.

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ST.-\NDING A N D JOVANOVICI-I: COBALT-60 7-RAYS 641

Measurements taken when first one and then the other of these photomulti- pliers acted as gating tube gave upper and lower limits on the ratio n,/nb.

T o minimize the effect of drifts in gain of the system with time, the various quantities entering the ratio were measured in rotation: the elastic scattering na; the corresponding backgrou~ld (with A1 in the beam); the direct radiation nb from Co60 (with C S ~ ~ ~ present); and the corresponding background (with Cs137 present). Individual ~neasure~uents usually lasted 5-10 minutes, during which time drift was negligible, and they were permuted to remove any possible effects of order. After a set of such measurements, the gating arrange- ments were interchanged and the measuren~ents repeated. This procedure was repeated until satisfactory statistical accuracy was attained, a determination of n,/nb a t one angle requiring several hundred individual measurements.

( 2 ) Resl~l t s Figure 11 shows the spectruiu of y-rays scattered from Pb a t 60' for various

gate settings, corresponding to the results of Fig. 4 obtained by the con- ventional method. Also shown in Fig. 11 is the spectrum of y-rays from the source b, for the same gate voltages as for the scattered radiation. If only elastic scattering were being observed, the spectrum of the scattered y-rays

PULSE HEIGHT (volts)

FIG. 11. Gated pulse-height spectra recorded in photomultiplier 1 for various gate voltage settings in photomultiplier 2 (actually the gating photomultiplier mas alternated, and the results averaged, as described in Section E(1)). a, ?-rays from the small Co60 source b a t the target position ( n b ) In each case the same curve IS fitted to the points. This curve, which was found to give the best average ht, is slightly asymmetrical, both sides being Gaussian, but with the width on the right about 5/6th of the width on the left side. 0, y-rays scattered from P b a t 60°, multiplied by 6.25 for ease of comparison (naX6.25). The gate voltage in- creased in going from A to D and the gate position is apparent from the number of pulses under the peak fib. In A, the gate accepted approxilnately all pulses fro111 the 1.33-Mev y-ray photopeak and a slllall number from 1.1'7-Mev y-rays, as shown by the tail on the low-energy side of the source spectrum, cf. Fig. 10. In B, the gate accepted =79% of the pulses from the 1.33-hiIev y-ray photopealr, in C ~ 5 8 % ) and in D =33y0.

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642 CANXDIAK JOURNAL OF PI-IYSICS. VOL. 40. 19F2

would, of course, have the same shape as the spectrum of the source b, so here there must be an appreciable contribution froin inelastic scattering a t the low gate settings. However, for higher gate voltages the spectra are almost identical, indicating removal of the inelastic component. Also the inelastic scattering appears mostly on the low-energy side of the spectra for all gates. Therefore na/nb, the ratio of scattered radiation to radiation fro111 the source, should approach its true value with increasing gate voltage and pulse height.

In order to see more clearly how the ratio na/nb behaves as the contribution of inelastic scattering is reduced, we have presented the ctata in a different way. The ratio was calculated separately for each gate setting and for each channel of the spectrum and the ratios plotted individually. The results for 60" scattering from Pb, obtained froin the data of Fig. 11, are shown in Fig. 12 and the results for scattering from Pb a t other angles in Figs. 13 to 1s. Table I1 gives the conditions under which the data were taken. In this table the "acceptance" is the percentage of the Co6" photopeak above the gate; where results for inore than one gate position are given, the flattest curve

PULSE HEIGHT (Volts) 30 35

PULSE HEIGHT (Volts)

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STANDING AND JOVASOVICII: COB:\LT-(I0 7-R.-\YS Gq3

P U L S E HEIGHT (Vo l t s ) P U L S E H E I G H T (Vol ts)

P U L S E HEIGHT (Volts)

FIGS. 12-18. Ratio curves for scattering of 1.33-Mev y-rays from lead. I n the highest channels where the ratios for various gates agreed within statistics, they have been combined and the combined result plotted. Statistical standard deviations in the esperimental points are denoted by bars. The curves are those calculated as described in Section E(2). I n Fig. 12 the lettered curves correspond to the curves with the same labels in Fig. 11.

corresponds to the snlallest acceptance. All targets were 5.5 in.X6 in., with thickness as shown in the table, and the distances source-target and counter- target varied from 80 to 130 cin for different angles. The absorption given in the table is the absorption in the target, with variations caused by differ- ences in target thickness and in path length in the target.

Consider first the simplest case, 90" scattering, shown in Fig. 13. Here the ratio n,/nb is the same within statistics for all channels a t the gate voltage used, showing that the inelastic scattering has been removed from them all. The ratio to be used in calculatiilg the scattering cross section is then just the ratio obtained from the total counting rates for all channels. We may note that the statistical accuracy is best for the points a t channels 32 and

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644 C.4SADIAN J0URN.-\L O F PIIYSICS. VOL. 10, l D G 2

TABLE I1

Esperime~ltal results for scattering from lead

Inelastic cross section Target per mc? (mb/sterad)

Accepta~lce thick- (per- ness Absorption Bacl~ground I< L

Angle centage) (g/cm2) (percentage) (percentage) end point end point Limit x,/nb

14 0 0 , 0 7 9 4 ~ . 002 16 1 . 7 .06 .101+.003 12 19 2 .3 0 (dashed) . 1 5 5 f ,008 13 . 4 (solid) 10 12 14 .5 . 8 . 2 2 0 f ,015 20* 250 13 . 7 4 f .O6 14

*An additional source of background which is significant only a t 30°, and just barely significant there, is Compton scattering of the CoGo y-ray a t 2.18 Mev. Although this y-ray has a n intensity of only 1.2X 10-6 X the intensity of the 1.33-Mev y-ray (Wolfson 1956), its Compton scattering cross section is =330Xthe 1.33-Mev elastic cross section, and its energy after Compton scattering through 30' is 1.39 Mev, just above the elastic line. I t makes an appreciable contribution to the higher channels of the spectrum, particularly since it is cut down less by the gate than the elastic line. Corrections are negligible in the lower half of the spectrum but in the upper 11alf cor- rections for this y-ray ranging from 0.7% a t channel 33 to 10% in channel 36 have been subtracted from the points before plotting Fig. 1 G .

33, since they correspond to the peak of the spectrum. The points a t the sides have much poorer accuracy as their counting rates are smaller by more than an order of magnitude. Also room background is more ilnportant there, particularly for channel 36, because an appreciable part of its background comes from radiation of energy greater than 1.33 Mev, so it is not cut down much by the gate.

Kow consider again the 60" data, plotted in Fig. 12. Here the ratio n,/nb

approaches a limit on the high-energy side of the spectrunl for all gate settings, but is considerably above this value on the low-energy side. For higher gate voltages, the ratios on the low-energy side decrease, but are larger than the limit even for the highest gates, indicating that there is still some inelastic scattering recorded. However, this method of plotting the data is a sensitive test for low-energy coiltarnination and its intensity is very low here, as is evident by cornpariilg Figs. 11 and 12. Only a sillall part of the elastic scattering spectrum appears in the low channels, where the low-energy spectrum is still near its maximum, so even a small ainount of low-energy radiation shows up clearly. For example, the ratio a t channel 29 for the highest gate is -50Yo above its limiting value, but because this point represents less than 20jo of the total spectrum, its effect on the over-all ratio would be less than 10jo if it were included.

In order to obtain a better liinit for the ratio and more iilforlllatioll on inelastic scattering, we have calculated the shape of the ratio curve to be expected in each case; this inay be done from knowledge of the gate position and the spectrum of inelastic scattering. The gate position is silllply deter- mined by the proportioil of the CoGO spectrum registered in each case (the

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STANDING AND JOVANOVICM: COBALT-60 ?-RAYS 645

acceptance), but we have no prior knowledge of the inelastic scattering except a t GO0. At that angle illelastic scattering was assumed to have the same value relative to the elastic scattering as was found experiinentally for gold (Section D) up to the end point of scattering from the I< shell (full energy minus the K shell binding energy). Beyond that energy, the only inelastic scattering is that from tlie L and higher shells, estimated by Randles (R) to be less than about 1/6th of the total for large scattering angles. A much larger proportion of the L sliell scattering lies above the gate, however, so it may in fact be more important here than the I< shell scattering; e.g., for the highest gate a t GO0 the calculated L sliell scattering accepted by tlie gate is over twice the I< shell scattering accepted if tlie L scattering is 1/6th of the total. The L shell contribution to the ratio may, in principle, be distinguished from the I< shell contribution by its higher energy spectrum, which was assuined to have the same shape as the I< sheil inelastic spectrum relative to its end point. An indicatioil of L shell scattering has already appeared in curve C of Fig. 6, which has a small tail beyond the I< shell end point.

The dashed curves in Fig. 12 were calculated assuming zero inelastic scattering from the L shell and taking the same proportion of I< shell inelastic scattering to elastic as was required to fit eq. (8) to the data for gold (Section D). Clearly these curves are too low in the central region, a fault which may be remedied by adding some L shell scattering, since it contributes to higher

~ channels than the I< shell. The solid curves in the same figure include enough L sliell inelastic scattering to fit the experimental results in the central region, an amount approsimately equal to Randles' upper limit being required. The over-all fit to the data is probably as good as may be expected, considering the uncertainty in the gate position and in the shape of the inelastic spectrum.

At other angles, we simply found the intensity of the K and L inelastic components required to fit the experinlental data, assuming that these com- ponents had the same spectral shape as a t GO0. The curves calculated are given in Figs. 13 to 18 and the corresponding inelastic cross sections a t the K and L shell end points in Table 11. Where there were results for severaI gate positions a t a given angle, they were all fitted using the same cross section. The inelastic scattering is very small a t 90" (cf. 60" a t approxilnately the same gate position) and increases with decreasing angle; the K shell cross sections are plotted in Fig. 17, along with the predictions of the form factor theory, which sho~v the same qualitative behavior. These cross sections should be talcen only as a rough estimate since the curves are rather sensitive to the exact gate setting used in the calculation, and there is some ambiguity in findiiig the relative amounts of I< and L shell scattering. However, they a t least provide a yardstick for measuring the inelastic contribution and should give a reasonable estimate of its influence on the ratio limit. For scattering angles of 20" or greater, the ratios approach a limit with increasing gate voltage and pulse height, so there is not much inelastic contribution included in the highest points and the ratio limit is not very sensitive to our inelastic scattering estimate; only for 12" is i t critical.

As mentioned in the last section, we tried to prevent errors from shifts in

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646 C r l X h l ) l A N JOURS:\L OF PHYSICS. VOL. -10, 1962

the photomultiplier gain with changing current; the current was kept al~llost the same for the various measurenients a t a given angle and the two photo- nlultipliers used as gates (Fig. 1) had counting rate shifts in opposite directions, so they gave upper and lower limits on the ratio ?~,/nb. At all angles larger than 30°, the ratios fro111 the two photo~llultipliers agreed within statistical error, but for the highest gates a t 30° and 20°, and for 12", the results were just outside statistics, one of the photon~ultipliers consistently reading high and the other low. However, it was found earlier that the opposite shifts of the two photonlultipliers were almost equal in magnitude, so should tend to cancel in the average; this is supported by the fact that the linlit of the high gate average ratio for 30" and 20" agrees with the limit for the low gate where the effect is greatly reduced. In any case the variation is well within the error estimated, e.g., the high gates a t 20" differ by f 5 % from the average while our estimated error is f lOyo .

Background for the scattering measurements was talten with the y-ray beam striking an aluminum target which gave the sanle photomultiplier current as the lead target. For angles of 45" or less the aluminum target had approximately the same number of electrons as the lead, but a t larger angles the number of electrons in the alunlinum target had to be considerably greater, because of the difference in scattered spectra (Fig. 3). Calculated pileup was never more than a few per cent of the total scattering from lead in the region of the elastic peak, and was mostly removed in the alulllillum background subtraction. A correction was added to the cross section for the elastic scattering from alunlinum; it was significant only a t 75" and 90" where it amounted to -6%. The background depended on the channel and is given in Table I1 for channels 32 and 33, since the radiation scattered fro111 lead had its peak value there. Except for 12", lllost of the background observed with the alunlinum target was general room background and was always less than one quarter of the lead counting rate in these channels. At 12", however, the scattering from alunlinuln was nearly half that from lead, presulnably because the gate accepted some counts fro111 the main Conlpton peak, which a t this angle is only about 5% below the elastic line in energy. In all channels, background has been subtracted from the experimental points before plotting, and its effect included in the standard deviations shown in Figs. 12 to 18.

The limit of the ratio n,/nb nlay be affected by counting rate shifts, by errors in subtracting background, and in estimating the shape of the spectrum, particularly the amount of the L shell contribution. We have made allowance for these errors as well as statistical error in the ratios given in Table 11, where the estimated probable errors given are typically three to six times the statistical errors.

From the ratio na/nb and the ratio of sources b/a, the absolute elastic scattering cross section may be calculated, as shown in Section B, and the results are given in Table 111 and Fig. 19, together with those found by using the conveiltional method to measure n,/nb (Section C) and those published by other workers. Note that the error quoted in our measurements does not include the error ( f 5%) in measuring the source ratio, since it has

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STAXDING ;\XI) JOVANOVICII: COBALT-GO 7-K.-\YS 647

TABLE I11 Differential cross section for elastic scattering of 1.33-i\.Iev y-rays fro111 lead (mb/steracl)

Present results'

Angle Two photomultipliers Conventional Other resultst

133+22 a t 13"17" (b) 147+22 a t 11.5"-18.5" (c)

43+13 a t 17"-23" (a)

10.0+1.5 a t 27"-33" (b) 13+3 a t 26.5"-33.5" (c)

7 .0+1.7 a t 30"-38" (d) 3 .81 + . 76 a t 30"-40" (a)

8 . 3 3 ~ 1 . 5 a t 32"-48" ( c )

, 145f .029 a t 85"-95" ((1) .16 a t 86.5.-94.5. (b) . 2 0 f .O5 a t 75"-102" (d)

.15 i fi

. l7f .O5 a t 84.5"-95.5" (c)

. l 2 f .02 a t 101 .5"-108.5"(b) . 1 0 4 f .021 a t 108"-12'2" (a)

. 5 5 f . 1 a t 115"-125" (e)

.16 (f

. 1 2 f .03 a t 122°-1430 fd)

.12+. 04 i.d . 1 1 3 f .023 a t 128"-142" ( h )

(a) I-Iara. Banaigs. I:bcrhard. Goldzahl, and Mey (1958). (b) Bernstein and Mann (1058). (c) Mann (1956). ((1) XVilson (1953). ( e ) Davey (10.53). ( j ) Messelt and Storruste (1050). (2) Strickler (1953). (11) Goldzahl and Eberl~ard (1057). *In addiliov to the errors quoted, there is an error conrinp from the measurerrient of tlie source ratio b/a xvliicl~

affects all the points in the same way and which me estlrriate as *5%. t T h e rrieasurements i n references ( b ) , (c), ( d ) , and ( e ) have been publislled only in graphical form. so there

may be some inaccuracy in our estractio~i of the numbers from their graphs.

the sanle effect on all points, nioving them up or dowil together, so it seemed better to keep it separate. The ranges of scattering angles accepted by the counter are listed when given by other worl;ers; our corresponding range (full width a t half maximum) is 52.5" for scattering angles of 30" or greater and =2" for scattering angles of 12" and 20°, i.e. a t 20" scattering angle our range is 19"-21". Our estimated error in nleasuring the scattering angles is ~ 0 . 5 " .

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648 CANADIAN JOURNAL O F PHYSICS. VOL. 40, l9G2

- I I I 1 I I I I I I - A WILSON (1953) -

- -

- + DAVEY (1953) - v STRICKLER (1953) - x MANN (1956)

-

l 8 0 MESSELT 8 STORRUSTE (1956) o BERNSTEIN 8 M A N N (1958) - - o HARA ET AL. (1958)

- l PRESENT EXPERIMENT

-

e -

- e\ - -

- -

- - -

- -

- -

- - -

-

- -

+ -

- -

- + -

- - -

- -

-

- + - - - + -

- - -

- -

- 0 -

0 dV ,..... "".". - ..... 0" - - I I I I I I I I I

- - I

0' 15" 30" 45" 60" 75" 90' 105' 120' 135" 1 5 0 °

ANGLE OF SCATTERING

FIG. 19. Cross section for elastic scattering of 1.33-i\/Iev y-rays from Icad. For errors see Tablc 111. The curves are theoretical curves, calculated as described in Section F(2).

F. DISCUSSION

( 1 ) Elastic Scattering : Experimental Resz~lts The elastic scattering cross sections obtained by other workers and our-

selves are given in Table I11 and Fig. 19. I t is obvious a t once that large systematic errors must have been present in many of the measurements cited, since the cross sections differ from one another by factors of 2 or more, while most of the errors are stated to be ,<20y0 (Table 111). We shall discuss the possible errors in our own experiment, and point out where me thinl; trouble may have arisen in the others.

The illelastic background is probably responsible, directly or indirectly, for illost of the difficulties. From Figs. 11 to 18 it appears that the two-photo- multiplier method has been quite effective in reducing the inelastic baclt- ground and enabling a good estimate of its residual effects to be made, and we believe that i t is much more reliable here than the conveiltional method. A comparisoil between our values obtained by the two-photomultiplier method and the conventional method illustrates the error liltely to be encountered in the latter; a t 60°, according to our results, the "elastic" pealr contains

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STANDING AND JOVANOVICA: COBALT-GO y R A Y S 649

-20y0 inelastic scattering, so the assuinption that it does not contribute would give a cross section -25% too high. The size of the effect depends, of course, on the counter resolution in each experiment, but it is perhaps significant that the biggest discrepancies between the individual results in Table I11 appear in the neighborhood of 60°, where the inelastic bacl;ground is highest relative to the elastic scattering.

I t seeins probable, however, that the inelastic scattering is itself not big enough to account for these large discrepancies, and that its secondary effects are even inore serious, since, by destroying the distinctive shape of the elastic pealts, it prevents other sources of error from being recognized. \Vhen the valley below a peak is filled in by inelastic scattering, a coilveiltional experi- ment merely gives information about its steeply rising upper side. In principle this might suffice to detect background or shifts in position, but in practice, with the statistical accuracy limited, it is very difficult indeed to give by this method any objective estimate of the size and shape ol the elastic peal; and what inay lie beneath it.

In particular, a shift in the pulse height of the elastic peak during the experiment is liltely to be interpreted as a change in counting rate when only the upper side of the peal; is available for analysis. The error is parti- cularly serious because of the steepness of this upper side; for exan~ple, a lyO shift in peak positioil between the measurement of the scattered spectruin and the reference source in Fig. 4 could give an error in cross section -40%. A large number of separate measurements average out randoill shifts, but a systematic shift in position between the scattered spectrum and the reference spectruin is a more serious problem. In fact such a shift will generally occur, as pointed out earlier, because of the variation of pl~otoinultiplier gain with counting rate. In the two-photomultiplier method also, a shift in the peak produces an error by changing the effective gate position, so to remove the effect we have depended on keeping the photomultiplier currents constant. Fortunately the effectiveness of this precaution could be checked by our two photomultipliers with opposite shifts, and as mentioned previously, any remaining shift gave an error well within our estimate.

Returning to the question of bacl;ground, the custoillary procedure has been to subtract from the lead spectrum the spectrum observed when a target of a light element, such as aluininum, replaced the lead. This, of course, sub- tracts rooin background and it has been argued that it will also account for other effects such as pileup and counts from the tail of the n~ain Coinpton peak. However, because of the very different inelastic spectra for aluminum and lead (Fig. 3), it certainly ~nalces no allowance for inelastic scattering, and for the saine reason, the hope that it will correct for these other effects seems rather optimistic unless the bacltground is small compared with the true counts. In our own case the aluininum was inserted during the background runs primarily to lceep the photoinultiplier currents constant. Separate checks were made for pileup, and the tail of the main Coinptoil pealc was significant only a t 12'. Except for this angle, the aluminuin background observed was mostly rooin background, and was 520% of the total counts from lead (Table 11), so we believe that our background correction is fairly accurate.

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650 C:\NADWN JOURNAL OF PkIYSICS. VOL. 40, 1962

Corrections for target absorption ( 2 15y0 in Table 11), made using Colgate's (1952) absorption coefficients corrected for finite solid angle, introduced an error estimated as 20.5y0 in the cross sections. Absorption in the lead in front of the counter was the same both for the scattered y-rays and for those from the source b, and so is autolnatically corrected; the small difference in attenuation coefficient, caused by the fact that the scattering is not actualIj7 completely elastic (owing to nuclear recoil), gives a negligible effect, even for these larger absorber thiclcnesses.

All these errors affect the relative values of the cross sections a t different angles, but in addition there may be errors which move all the points up or down together, i f the y-ray flux a t the target is different from that estimated in the source strength ratio measurement. As discussed earlier, an error of 4 ~ 5 7 ~ has been estimated for that measurement; however, there could be additional errors if there were appreciable contributions to the y-ray flux a t the target which were not observed in the narrow bean1 source strength ratio measurement, slit scattering for example. Although scattering in the source and from the slits is large in total (Cormack and Johns 1958), only a very srnall fraction of it is accepted by the counter, and we estimate any corrections for these effects to be 2170. Also such effects might be expected to vary across the beam, but the scattering from a small target placed in various parts of the beam was found to be constant well within our estimated error.

Throughout this discussion the error we have referred to in our own measure- ments is our best estimate of the probable error, including systematic errors, i.e. we believe that the probability of the correct result lying within our estimated linlits is -~507~',.

Of the results in Table I11 and Fig. 19, only those of Hara et al . (1958) are in reasonable agreement with ours within their stated errors; the cross sections of Bernstein and Mann (1958) and the earlier experinlents are considerably larger. LlIost of the errors discussed above tend to male the measured cross sections too large, so this is perhaps not surprising.

( 2 ) Elastic Scattering: Comparison with Tlzeory From Table I it is seen that our experilnental cross sections for elastic

scattering of 1.33-Mev y-rays from aluminum and copper a t 90' agree with the nuclear Thonzson cross sections, in agreement with the predictions of field theory and the experiments of Alvarez et al. (1958) a t 124' and 1.G Mev. For tin the contribution of Rayleigh scattering becomes important but still our cross section agrees with theory within experinlental error.

For lead Rayleigh scattering gives the largest contribution even a t 90' and predominates a t small angles, so its calculation determines the accuracy of the theoretical cross section. As nlentioned earlier, the form factor approxi- mation was shown to be very poor by the accurate calculations of Brown and Mayers (1957), who computed the amplitudes for the scattering of 2.56 1nc2 y-rays fro111 the R shell of mercury. They found that the anlplitude for scattering with y-ray polarization change was given fairly accurately by the form factor, but that the amplitude for scattering without polarization change was ~nuch smaller than the for111 factor prediction. For the latter case they

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STANDING A N D JOVANOVICH: COBALT-60 7-RAYS 65 L

gave a semi-empirical "corrected for111 factor" which agreed with the exact calculatioils a t small angles. We have calculated the ainplitudes for scattering of 1.33-Mev y-rays froin the I< shell of lead by illultiplyiilg Brown and Mal~ers' amplitude for polarization change by the ratio of the forin factor ainplitudes (Pb a t 1.332 NIev/Hg a t 2.56 111c3) and inultiplyiilg their amplitude for no polarization change by the ratio of the "corrected form factor" aillplitudes (for angles <60°); this was done a t each angle and gave corrections to the mercury cross section 5 +lo%.

Unfortunately the scattering from the L shell has not been calculated accurately. In line with their K shell results Brown and Mayers suggested using the forin factor approximation for the L shell scattering alnplitude with polarization change. For the a~nplitude without polarizatio~l change even their "corrected forin factor" gave reasonable results for the I< shell only a t sinall angles, where the mon~entum transfer from the y-ray was smaller than the K shell electron momentum; for larger momentum transfers the exact calcu- lations gave ainplitudes much smaller than this, e.g. by a factor -3 a t 60". Since the momentum of the electrons in the L shell is less than half that in the K shell, this decrease might be expected to set in a t correspondingly smaller angles, so Brown and Mayers suggested that it is better to neglect altogether the L shell scattering amplitude without polarization change than to use the "corrected form factor".

The solid line in Fig. 19 is the theoretical cross section including contribu- tions from nuclear Thoinson scattering, Rayleigh scattering from the R shell (Brown and Mayers' results corrected for Z and energy as described above), and Rayleigh scattering from the L shell (zero amplitude for no polarization change and the form factor amplitude for polarizatioil change); this corre- sponds to Brown and NIayers' suggestions and would seen1 to be the best theoretical estimate a t present. The dashed curve above it includes also the L shell ainplitude for no polarization change calculated from the "corrected form factor", and the dotted curve below is calculated for no L shell scattering whatever, so they would seem to be reasonable upper and lower lilnits to the theoretical cross section. We have inade rough estimates of the corrections to these cross sections from nuclear resonant scattering (Levinger 1951) and from the effects of the outer electrons using Mayers' (1957) wave functions for mercury; since the corrections are 52y0 , well within the uncertainty of the Rayleigh scattering calculations, we have neglected them.

For angles less than 90°, our experimental results are in excellent agreement with the predictions of Brown and NIayers, shown in the solid curve. At larger angles the points fall systematically below the solid curve, but still lie above the dotted one except for the point a t 150° which is just below it. We therefore conclude that our experimental cross sectioil for the scattering of 1.33-Mev y-rays froin lead agrees with that predicted theoretically for Rayleigh and nuclear Thornson scattering alone, within the present ullcertaillt~~ of the calculations, so there is no evidence from our measurements for the existence of Delbriick scattering. An upper limit to its intensity is now set by the uncertainty in the theory, since we believe that our experiinental cross section

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is considerably inore accurate than the calculations; i f the dashed curve is taken as the upper limit of the theoretical Rayleigh and nuclear Thomson cross section our experimental points give upper linlits on the Delbruck scattering amplitude 5aT/2 for angles >,60°, -aT a t Go, -2.5aT a t 30°, ancl -SaT a t 20°, where aT is the nuclear Thomson scattering amplitude.

The imaginary part of the Delbruclc scattering amplitude has been calcu- lated a t 2.G2 Mev and 6.14 Mev (Kessler 1958; Zernilc 1960), but extrapolation to 1.33 AiIev suggests its value is small there, giving a contribution to the cross section certainly 5 +2% and probably much less. The theoretical value for the real part of the Delbruclc scattering amplitude is known accurately only a t oO, where it is -5aT for 1.33-Mev y-rays on lead, about 40 tiines as large as the imaginary part (Rohrlich and Glucltstern 1952; Toll 1952; Rohr- lich 1957). The extensions to finite angles, which have been made (Toll 1952; Bethe and Rohrlich 1952; Werntz and Cheston 1957), are not really valid for our energy and scattering angles but in any case would predict an alnplit~ide no greater than the lilnits given in the last paragraph. Therefore, in spite of the fact that we find no positive evidence for the presence of Delbruclc scatter- ing in our measurements, we cannot claiin ally contradiction with theory. Improvement of this situation will require better calculations, both of the I; shell Rayleigh scattering and of Delbruclc scattering.

AIoffat and Stringfellow (1960) have observed the scattering of 87-Mev y-ra>.s a t small angles, and obtain results consistent with the predictions of Bethe ancl Rohrlich (1952); however, since the imaginary part of the scattering amplitude predominates a t that energy, their results unfortunately give no information about vacuum polarization. At 2.62 Mev the present experimental results (Eberhard et al. 1958; Bernstein and i\/Iann 1958) are considerably larger than the theoretical cross section for Rayleigh and nuclear Thorns011 scattering (Goldzahl et a/ . 1959), and the difference is larger than can be accountecl for- by the imaginary part of the Delbruclc scattering amplitude, so it would be interesting to have inore measurements and calculatiolls in that neighborhoocl.

ACKNOWLEDGMENTS

We are indebted to Professor R. Sherr and Dr. R. W. Peelle for originally arousing our interest in this problem. Financial support for the work was provided b>- the Kational Research Council of Canada and the coinputations a t Washington University were carried out with the assistance of the U.S. Air Force Office of Scientific Research, Air Research and Developinent Coin- mand. \Ve are particularly grateful to the Manitoba Cancer Foundation and to its director, Dr. R. J . Walton, for permission to use its CoGO beam therapy unit. We thanl; Dr. A. Bernstein, Dr. G. E. Brown, Dr. L. Goldzahl, Dr. A. I<. kIann, Dr. J. M. Motz, Mr. J. Randles, and Dr. A. Storruste for inforinative discussions and for collllllunication of their results before publi- cation, and hiIiss P. Barnes, Mr. T. Griffith, and especially Mr. G. Hardie for valuable assistance.

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